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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Seek another solution of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_12.html ./knowl/eq7_13.html">
\begin{equation}
{\bf x}^{(2)}=t \vec{\xi} e^{r_1 t}=t \vec{\xi},\tag{6.4.7}
\end{equation}
</div>
<p class="continuation">where <span class="process-math">\(\vec{\xi}\)</span> is to be determined. Substituting (<a href="" class="xref" data-knowl="./knowl/eq7_12.html" title="Equation 6.4.7">(6.4.7)</a>) into (<a href="" class="xref" data-knowl="./knowl/eq7_13.html" title="Equation 6.4.4">(6.4.4)</a>), we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_12.html ./knowl/eq7_13.html">
\begin{equation*}
\vec{\xi}=\left(
\begin{array}{cc}
4 &amp; -2\\
8 &amp; -4
\end{array}
\right) t \vec{\xi}.
\end{equation*}
</div>
<p class="continuation">The left hand side is irrelevant with <span class="process-math">\(t\)</span> while the right hand side is and this can not be satisfied except <span class="process-math">\(\vec{\xi}={\bf 0}\text{.}\)</span></p>
<span class="incontext"><a href="sec6_4.html#p-274" class="internal">in-context</a></span>
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